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.. index::
single: Affine Variational Inequalities (AVI)
.. contents::
.. _avi_problem:
Affine Variational Inequalities (AVI)
*************************************
Problem statement
=================
The Affine Variational Inequality (AVI) is defined by
Given :math:`q\in\mathbb{R}^n` , :math:`M\in\mathbb{R}^{n\times n}` and a set :math:`K\in\mathbb{R}^n` , find :math:`z\in\mathbb{R}^n` such that:
.. math::
\begin{equation*}(Mz+q)^T(y -z) \geq 0,\quad \text{ for all } y \in K,\end{equation*}
or equivalently,
.. math::
\begin{equation*}-(Mz + q) \in \mathcal{N}_K(z)\end{equation*}
where :math:`\mathcal{N}_K` is the normal cone to :math:`K` at :math:`z` .
The AVI is a special case of a Variational Inequality (VI), where the function :math:`F` is affine. For VI solvers, see :ref:`vi_problem`.
From more details on theory and analysis of AVI (and VI in general), we refer to :cite:`Facchinei.2003`
Implementation in numerics
==========================
Structure to define the problem: :class:`AffineVariationalInequalities`.
The generic driver for all VI is :func:`avi_driver()`.
solvers list :enum:`AVI_SOLVER`
.. _avi_solvers:
AVI Available solvers
=====================
Cao Ferris (:enumerator:`SICONOS_AVI_CAOFERRIS`)
------------------------------------------------
Direct solver for AVI based on pivoting method principle for degenerate problem. Choice of pivot variable is performed via lexicographic ordering
driver: :func:`avi_caoferris()`
parameters:
* iparam[SICONOS_IPARAM_MAX_ITER] = 10000
* dparam[SICONOS_DPARAM_TOL] = 1e-12
Pathvi (:enumerator:`SICONOS_AVI_PATHAVI`)
------------------------------------------
Direct solver for VI based on pivoting method principle for degenerate problem.
Ref: "A structure-preserving Pivotal Method for Affine Variational Inequalities" Y. Kim, O. Huber, M.C. Ferris, Math Prog B (2017).
driver: :func:`avi_pathvi()`
parameters:
* iparam[SICONOS_IPARAM_MAX_ITER] = 10000
* dparam[SICONOS_DPARAM_TOL] = 1e-12
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